Random Variables and Measurable Functions

Abstract

In statistical work, the basic variables of interest are random variables. Despite the terminology, random variables have no inherent relationship to any intuitive concept of randomness or chance. Instead, random variables are functions on which parameters of interest are readily defined. In consonance with practice since Kolmogorov (1933), random variables are defined as special cases of measurable functions. In turn, definitions of measurable functions used in this chapter are based on the definition of real measurable functions in Daniell (1920). In Section 3.1, Daniell’s definition of33 real measurable functions is used to define real random variables and random vectors, and basic properties of measurable functions are developed. In Section 3.2, the relationship between regular Daniell integrals and continuous transformations is developed, and applications of real Baire functions are considered. In Section 3.3, summary measures based on intervals are used to describe measurable functions. Basic properties of these measures are discussed, and applications to description of data are presented. These measures are shown to provide basic characterizations of distributions.